Том 23, № 1 (2018)

A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.

For any strictly convex planar domain Ω ⊂ R² with a C^∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and \(\bar \Omega \) with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sⁿ}_n≥1 (resp. \({\left\{ {{{\bar S}^n}} \right\}_{n \geqslant 1}}\)) of period going to infinity such that Sⁿ and \({\bar S^n}\) have the same period and perimeter for each n.

In this paper we consider a symmetric restricted circular three-body problem on the surface S² of constant Gaussian curvature κ = 1. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius a ∈ (0, 1). It is verified that both poles of S² are equilibrium points for any value of the parameter a. This problem is modeled through a Hamiltonian system of two degrees of freedom depending on the parameter a. Using results concerning nonlinear stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances. It is verified that for the north pole there are two values of bifurcation (on the stability) \(a = \frac{{\sqrt {4 - \sqrt 2 } }}{2}\) and \(a = \sqrt {\frac{2}{3}} \), while the south pole has one value of bifurcation \(a = \frac{{\sqrt 3 }}{2}\).

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.